Kidney. As long as there is one functioning kidney, you do not need a transplant. After all, kidney transplant recipients ultimately only have one functioning kidney.
No. The remaining kidney (assuming it has not also been damaged) can usually take over and there will be no perceptible change. If you have a single kidney you need to be extra careful to protect it (avoid contact sports, use seat belts, avoid excessive use of some medications like aspirin, ibuprofen, aleve). Be sure too check you kidney function yearly to make sure your kidney remains healthy.
Generally no. Although it can depend on the cause of the kidney failure, in most cases, if one kidney fails, there is sufficient reserve in the other kidney that you can survive with just the one remaining kidney.